在Bernal双层石墨烯（BBG）中，垂直位移场使导带底部变平，从而促进低电子密度下强关联电子态的形成。这里，我们关注在足够大的位移场、低电子密度和低温的特定区域出现的维格纳晶体（WC）态。我们首先考虑没有三角翘曲的BBG模型，理论上表明贝里曲率导致一种新的WC态，其中当位移场超过临界值时，电子获得自发轨道磁化。然后，我们考虑BBG中三角翘曲的影响，并表明它们导致WC相随密度变化的异常“双重返入”行为。与三角翘曲相关的旋转对称性破缺导致WC态中非平凡的“迷你谷序”，该序在位移场临界值处突然变化。在两种情况下，我们根据密度、位移场和温度估计了WC态的相边界。本文补充了我们最近的工作[Joy et al., Phys. Rev. Lett. 135, 256502 (2025)]。
英文摘要：
In Bernal bilayer graphene (BBG), a perpendicular displacement field flattens the bottom of the conduction band and thereby facilitates the formation of strongly correlated electron states at low electron density. Here, we focus on the Wigner crystal (WC) state, which appears in a certain regime of sufficiently large displacement field, low electron density, and low temperature. We first consider a model of BBG without trigonal warping, and we show theoretically that Berry curvature leads to a new kind of WC state in which the electrons acquire a spontaneous orbital magnetization when the displacement field exceeds a critical value. We then consider the effects of trigonal warping in BBG, and we show that they lead to an unusual ``doubly re-entrant" behavior of the WC phase as a function of density. The rotational symmetry breaking associated with trigonal warping leads to a nontrivial ``minivalley order" in the WC state, which changes abruptly at a critical value of displacement field. In both cases, we estimate the phase boundary of the WC state in terms of density, displacement field, and temperature. This paper is complementary to our recent work [Joy et al., Phys. Rev. Lett. 135, 256502 (2025)].
We investigate the effects of coupling between the two zigzag edges of graphene nanoribbons on the Wigner crystallization of electrons and holes using a combination of tight-binding, mean field Hubbard and many-body configuration interaction methods. We show that the thickness of the nanoribbon plays a crucial role in the formation of Wigner crystal. For ribbon widths smaller than 16 \mboxÅ, increased kinetic energy overcomes the long-range Coulomb repulsion and suppresses the Wigner crystallization. For wider ribbons up to 38 \mboxÅ wide, strong Wigner localization is observed for even number…
We consider the fate of the Wigner crystal state in a two dimensional system of massive Dirac electrons as the effective fine structure constant $α$ is increased. In a Dirac system, larger $α$ naively corresponds to stronger electron-electron interactions, but it also implies a stronger interband dielectric response that effectively renormalizes the electron charge. We calculate the critical density and critical temperature associated with quantum and thermal melting of the Wigner crystal state using two independent approaches. We show that at $α\gg 1$, the Wigner crystal state is best underst…
In a certain regime of low carrier densities and strong correlations, electrons can crystallize into a periodic arrangement of charge known as Wigner crystal. Such phases are particularly interesting in one dimension (1D) as they display a variety of charge and spin ground states which may be harnessed in quantum devices as high-fidelity transmitters of spin information. Recent theoretical studies suggest that the strong Coulomb interactions in Mott insulators and other flat band systems, may provide an attractive higher temperature platform for Wigner crystallization, but due to materials and…
We study the Wigner crystallization on partially filled topological flat bands. We identify the Wigner crystals by analyzing the cartesian and angular Fourier transform of the pair correlation density of the many-body ground state obtained using exact diagonalization. The crystallization strength measured by the magnitude of the Fourier peaks, increases with decreasing particle density. The shape of the resulting Wigner crystals is determined by the boundary conditions of the chosen plaquette and to a large extent independent on the underlying lattice, including its topology, and follows the b…
We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $ν=2,3$ charges occupy a unit moiré cell, a feature attributed to Mottness, but not for $ν=1$, clearly inconsistent with Mott insulation. We compute $r_s=E_U/E_K$, where $E_U$ and $E_K$ are the potential and kinetic energies, respectively, and show that (i) the Mott criterion lies at a density $10^4$ higher than in the experiments and (ii) a transition to a series of Wigner crystalline states exists as a function of $ν$. We find, for $ν=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the…
We introduce a new approach for the correlation energy of one- and two-valley two-dimensional electron gas (2DEG) systems. Our approach is based on a random phase approximation at high densities and a classical approach at low densities, with interpolation between the two limits. This approach gives excellent agreement with available Quantum Monte Carlo (QMC) calculations. We employ the two-valley 2DEG model to describe the electron correlations in monolayer transition metal dichalcogenides (TMDs). The zero-temperature transition from a Fermi liquid to a quantum Wigner crystal phase in monolay…
Critical gate distance for Wigner crystallization in the two-dimensional electron gas
Center for Computational Quantum Physics, Flatiron Institute（Flatiron Institute 计算量子物理中心）；Department of Physics, Stanford University（斯坦福大学物理系）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Department of Physics and Astronomy, Hofstra University（霍夫斯特拉大学物理与天文系）；Department of Physics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校物理系）
A system of confined charged electrons interacting via the long-range Coulomb force can form a Wigner crystal due to their mutual repulsion. This happens when the potential energy of the system dominates over its kinetic energy, i.e., at low temperatures for a classical system and at low densities for a quantum one. At $T=0$, the system is governed by quantum mechanics, and hence, the spatial density peaks associated with crystalline charge localization are sharpened for a lower average density. Conversely, in the classical limit of high temperatures, the crystalline spatial density peaks are …
The competition between kinetic energy and Coulomb interactions in electronic systems can lead to complex many-body ground states with competing superconducting, charge density wave, and magnetic orders. Here we study the low temperature phases of a strongly interacting zinc-oxide-based high mobility two dimensional electron system that displays a tunable metal-insulator transition. Through a comprehensive analysis of the dependence of electronic transport on temperature, carrier density, in-plane and perpendicular magnetic fields, and voltage bias, we provide evidence for the existence of com…
The flat band has attracted a lot of attention because it gives rise to many exotic phases, as recently demonstrated in magic angle twisted bilayer graphene. Here, based on first-principles calculations, we identify a metal-insulator transition in boron triangular Kagome lattice with a spin-polarized flat band at 2/3-filling. This phase transition is accompanied by the formation of a Wigner crystal, which is driven by Fermi surface nesting effect and thereby strong electron-phonon interactions, keeping ferromagnetism. Our calculation results suggest that boron triangular Kagome lattices with p…
We study fermions in two dimensions interacting via a long-ranged 1/r potential for small particle separations and a short-ranged 1/r^3 potential for larger separations in comparison to a length scale ξ. We compute the energy of the Wigner crystal and of the homogeneous Fermi liquid phases using a variational approach, and determined the phase diagram as a function of density and ξat zero temperature. We discuss the collective modes in the Fermi liquid phase, finite temperature effects on the phase diagram, and possible experimental realizations of this model.
Recent experiments on the twisted transition metal dichalcogenide (TMD) material, $\rm WSe_2/WS_2$, have observed insulating states at fractional occupancy of the moiré bands. Such states were conceived as generalized Wigner crystals (GWCs). In this article, we investigate the problem of Wigner crystallization in the presence of an underlying (moiré) lattice. Based on the best estimates of the system parameters, we find a variety of homobilayer and heterobilayer TMDs to be excellent candidates for realizing GWCs. In particular, our analysis based on $r_{s}$ indicates that $\rm MoSe_{2}$ (among…
The existence of Wigner crystallization, one of the most significant hallmarks of strong electron correlations, has to date only been definitively observed in two-dimensional systems. In one-dimensional (1D) quantum wires Wigner crystals correspond to regularly spaced electrons; however, weakening the confinement and allowing the electrons to relax in a second dimension is predicted to lead to the formation of a new ground state constituting a zigzag chain with nontrivial spin phases and properties. Here we report the observation of such zigzag Wigner crystals by use of on-chip charge and spin…
How the two dimensional (2D) quantum Wigner crystal (WC) transforms into the metallic liquid phase remains to be an outstanding problem in physics. In theories considering the 2D WC to liquid transition in the clean limit, it was suggested that a number of intermediate phases might exist. We have studied the transformation between the metallic fluid phase and the low magnetic field reentrant insulating phase (RIP) which was interpreted as due to WC formation [Qiu et al, PRL 108, 106404 (2012)], in a strongly correlated 2D hole system with large interaction parameter $r_s$ ($\sim~$20-30) and hi…
Interaction in a flat band is magnified due to the divergence in the density of states, which gives rise to a variety of many-body phenomena such as ferromagnetism and Wigner crystallization. Until now, however, most studies of the flat band physics are based on model systems, making their experimental realization a distant future. Here, we propose a class of systems made of real atoms, namely, carbon atoms with realistic physical interactions (dubbed here as Kagome graphene/graphyne). Density functional theory calculations reveal that these Kagome lattices offer a controllable way to realize …
We study Wigner crystallization of electron systems in phosphorene quantum dots with confinement of an electrostatic origin with both circular and elongated geometry. The anisotropy of the effective mass allows for the formation of Wigner molecules in the laboratory frame with a confined charge density that has lower symmetry than the confinement potential. We find that in circular quantum dots separate single-electron islands are formed for two and four confined electrons but not for three trapped carriers. The spectral signatures of the Wigner crystallization to be resolved by transport spec…
Dynamic Moiré Potentials and Robust Wigner Crystallization in Large-Scale Twisted Transition Metal Dichalcogenides
Hefei National Research Center for Physical Sciences at the Microscale, and Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家物理科学（微观）研究中心和合肥国家实验室）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部研究生院）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；State Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China（中国科学技术大学精密与智能化学国家重点实验室、化学物理系和安徽应用数学中心）
Chiral Wigner crystal phases induced by Berry curvature
Ohio State University（俄亥俄州立大学）；National High Magnetic Field Laboratory（国家高磁场实验室）；Florida State University（佛罗里达州立大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider temperature-induced melting of a Wigner solid in one dimensional (1D) and two dimensional (2D) lattices of electrons interacting via the long-range Coulomb interaction in the presence of strong disorder arising from charged impurities in the system. The system simulates semiconductor-based 2D electron layers where Wigner crystallization is often claimed to be observed experimentally. Using exact diagonalization and utilizing the inverse participation ratio as well as conductance to distinguish between the localized insulating solid phase and the extended metallic liquid phase, we f…
The dominance of interactions over kinetic energy lies at the heart of strongly correlated quantum matter, from fractional quantum Hall liquids, to atoms in optical lattices and twisted bilayer graphene. Crystalline phases often compete with correlated quantum liquids, and transitions between them occur when the energy cost of forming a density wave approaches zero. A prime example occurs for electrons in high magnetic fields, where the instability of quantum Hall liquids towards a Wigner crystal is heralded by a roton-like softening of density modulations at the magnetic length. Remarkably, i…
Visualizing the Impact of Quenched Disorder on 2D Electron Wigner Solids
University of California, Berkeley（加州大学伯克利分校）；Flatiron Institute（平顿研究所）；University of New Mexico（新墨西哥大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Hofstra University（霍夫斯特拉大学）；Arizona State University（亚利桑那州立大学）；National Institute for Materials Science（日本国立材料研究所）
Recent experiments [arXiv: 1808.07865] on twisted bilayer graphene (TBLG) show that under hydrostatic pressure, an insulating state at quarter-filling of the moiré superlattice (i.e., one charge per supercell) emerges, in sharp contrast with the previous ambient pressure measurements of Cao et al. where the quarter--filling state (QFS) is a metal [Nature 556, 43 & 80 (2018)]. In fact, the insulating state at the other commensurate fillings of two and three charges per supercell is also enhanced under applied pressure. Based on realistic computations of the band structure for TBLG which show th…
Measuring the magnetoresistance (MR) of ultraclean {\it GaAs} two-dimensional holes in a large $r_s$ range of 20-50, two striking behaviors in relation to the spin-orbit coupling (SOC) emerge in response to strong electron-electron interaction. First, in exact correspondence to the zero-field metal-to-insulator transition (MIT), the sign of the MR switches from being positive in the metallic regime to being negative in the insulating regime when the carrier density crosses the critical density $p_c$ of MIT ($r_s\sim 39$). Second, as the SOC-driven correction $Δρ$ to the MR decreases with reduc…
Spin and Valley dependent analysis of the two-dimensional low-density electron system in Si-MOSFETS
硅金属氧化物场效应晶体管（MOSFET）中的二维电子系统（2DES）由两种不同的电子流体相互作用组成。我们计算的总能量作为密度的函数$n$，和自旋极化$ζ$在强相关的低密度制度，使用一个经典的映射到一个超网链（CHNC）方程包括桥项。在这里，十个分布函数，所产生的自旋和谷指数，自洽计算，以获得总自由能，化学势，压缩性和自旋磁化率。将T=0的结果与Conti等人的2谷量子蒙特卡罗（QMC）数据（T=0时，$ζ=0$）进行了比较，发现两者非常吻合。然而，与单谷2DES不同，它表明{ \it非偏振相总是2谷系统中的稳定相}，直到在$r_s=42$处的维格纳结晶。这导致$g^*$对自旋极化和密度不敏感。从自由能计算的压缩性和自旋磁化率增强证实了一个简单的方法的有效性的耦合模式形成的基础上的两个谷响应。这三种方法，QMC，CHNC和耦合模理论是一致的。我们的结果不包含{ \it ad hoc}拟合参数。他们同意实验，并没有援引杂质效应或金属-绝缘体过渡现象。
